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Date	November 2021	Marks available	5	Reference code	21N.2.SL.TZ0.5
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Show that	Question number	5	Adapted from	N/A

Question

The following diagram shows a semicircle with centre $O$ and radius $r$ . Points $P, Q$ and $R$ lie on the circumference of the circle, such that $PQ = 2 r$ and $R \hat{O} Q = θ$ , where $0 < θ < π$ .

Given that the areas of the two shaded regions are equal, show that $θ = 2 \sin θ$ .

[5]

Hence determine the value of $θ$ .

[1]

Markscheme

attempt to find the area of either shaded region in terms of $r$ and $θ$ (M1)

Note: Do not award M1 if they have only copied from the booklet and not applied to the shaded area.

Area of segment $= \frac{1}{2} r^{2} θ - \frac{1}{2} r^{2} \sin θ$ A1

Area of triangle $= \frac{1}{2} r^{2} \sin (π - θ)$ A1

correct equation in terms of $θ$ only (A1)

$θ - \sin θ = \sin (π - θ)$

$θ - \sin θ = \sin θ$ A1

$θ = 2 \sin θ$ AG

Note: Award a maximum of M1A1A0A0A0 if a candidate uses degrees (i.e., $\frac{1}{2} r^{2} \sin (180 ° - θ)$ ), even if later work is correct.

Note: If a candidate directly states that the area of the triangle is $\frac{1}{2} r^{2} \sin θ$ , award a maximum of M1A1A0A1A1.

[5 marks]

$θ = 1.89549 \dots$

$θ = 1.90$ A1

Note: Award A0 if there is more than one solution. Award A0 for an answer in degrees.

[1 mark]

Examiners report

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.2—2d and 3d trig, sine rule, cosine rule, area

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19M.1.SL.TZ1.T_8a:
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18M.1.SL.TZ1.T_8c:
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19M.2.SL.TZ2.S_3b:
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17M.1.SL.TZ2.T_3b:
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19N.2.SL.TZ0.S_6:
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19N.1.SL.TZ0.S_9a.ii:
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22M.2.SL.TZ1.7b:
Given that $A \hat{V} B = 40 °$ , find $AB$ .
22M.2.SL.TZ2.7b:
Find the area of this piece of land.
22M.2.SL.TZ2.7c:
Find $CÂB$ .
22M.2.SL.TZ2.7d:
Find the distance from point $B$ to point $D$ .
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Show that angle $EDC = 48.0 °$ , correct to three significant figures.
21M.2.SL.TZ1.8e.ii:
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18M.1.SL.TZ2.T_8a:
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SPM.1.SL.TZ0.1b:
Find the area of triangle ABC.
18N.2.SL.TZ0.S_7a:
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16N.2.SL.TZ0.T_5d:
the area of quadrilateral ABCD;
21M.2.SL.TZ1.8a:
Write down an expression for $r$ in terms of $θ$ .
21M.2.SL.TZ1.8e.i:
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21M.3.AHL.TZ1.2c:
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21M.3.AHL.TZ1.2d:
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17M.2.AHL.TZ2.H_4a:
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18M.1.SL.TZ1.T_8b:
Find the size of angle APB.
19N.1.SL.TZ0.S_9b:
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21M.2.SL.TZ1.8b:
Show that the area of the field that the horse can reach is $\frac{392}{θ^{2}} (θ + \sin θ)$ .
19M.1.AHL.TZ1.H_4a:
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19N.1.AHL.TZ0.H_9c.i:
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EXN.2.SL.TZ0.2b:
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17N.2.SL.TZ0.S_1a:
Find BC.
SPM.2.SL.TZ0.7c:
Find ${\text{B}}\mathop {\text{C}}\limits^ \wedge {\text{A}}$ .
21M.2.SL.TZ1.8d:
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SPM.2.SL.TZ0.7d:
Find the bearing that Jacob must take to point C.
EXN.1.AHL.TZ0.7:
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Find an expression for $PS$ in terms of $x, y, \sin β$ and $\sin (α + β)$ .
22M.2.SL.TZ2.1a:
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21N.2.SL.TZ0.5b:
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22M.2.SL.TZ2.7a:
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18M.2.SL.TZ2.T_5c:
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17N.1.SL.TZ0.T_10c:
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19M.1.SL.TZ2.T_9b:
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16N.2.SL.TZ0.T_5f:
the length of the fence, BP.
17M.2.SL.TZ1.T_2a.i:
Write down an equation for the area of ABCDE using the above information.
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19M.1.SL.TZ1.T_10b:
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19M.1.SL.TZ1.S_3a:
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19M.1.SL.TZ1.S_3b:
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19M.2.AHL.TZ2.H_1:
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Find all the interior angles of the triangle. Give your answers in degrees to one decimal place.
18N.2.SL.TZ0.T_5b:
Calculate the area of triangle ABC.
19N.1.AHL.TZ0.H_9c.ii:
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17N.1.SL.TZ0.S_4a:
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17M.1.SL.TZ2.T_3a:
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19M.1.SL.TZ1.S_3c:
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19N.1.AHL.TZ0.H_9a:
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19N.1.SL.TZ0.S_9c:
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17M.1.SL.TZ1.S_3:
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Find PR.
19M.1.AHL.TZ1.H_4b:
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17N.1.SL.TZ0.S_4b:
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18M.2.SL.TZ2.S_2a:
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18M.2.SL.TZ2.S_2b:
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EXN.2.SL.TZ0.2a:
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18N.2.SL.TZ0.S_7b:
Hence or otherwise, find the total distance along the road where the signal from the tower can reach cellular phones.
21M.1.SL.TZ2.6:
The following diagram shows triangle $ABC$ , with $AB = 10$ , $BC = x$ and $AC = 2 x$ .

Given that $\cos \hat{C} = \frac{3}{4}$ , find the area of the triangle.

Give your answer in the form $\frac{p \sqrt{q}}{2}$ where $p, q \in ℤ^{+}$ .
21M.2.SL.TZ1.8c:
The area of field that the horse can reach is $460 m^{2}$ . Find the value of $θ$ .
21N.2.SL.TZ0.3:
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Find the smallest possible perimeter of triangle $ABC$ .
17M.2.SL.TZ2.T_4e:
Calculate the total length of the track.
17M.1.SL.TZ1.T_13a:
Find the length of BC.
18N.2.SL.TZ0.T_5c:
Find the length of AC.
16N.2.SL.TZ0.T_5b:
the size of angle DAB;
19M.1.SL.TZ2.T_9a:
Find the angle of elevation from B to C.
17M.2.SL.TZ2.T_4b:
Show that angle DBC is 48.7°, correct to three significant figures.
17M.1.SL.TZ1.T_13b:
Find the area of the postage stamp.
20N.2.SL.TZ0.T_3d:
Pedro draws a circle, with centre at point $E$ , passing through point $C$ . Part of the circle is shown in the diagram.

Show that point $A$ lies outside this circle. Justify your reasoning.
17M.2.SL.TZ1.T_2c:
Show that angle ${\rm{B\hat AE}} = 67.4^\circ$ , correct to one decimal place.
17M.2.SL.TZ2.T_4d:
Find the length of the path CE.
16N.2.SL.TZ0.T_5a:
the length of BD;
19M.2.SL.TZ2.T_2d:
The length of path AD is 287 m.

Find the area of the region ABCD.
16N.1.SL.TZ0.T_11a:
Calculate CD, the height of the observation deck above the ground.
19M.1.SL.TZ1.T_10a:
Find the size, in degrees, of angle BÂC.
18M.2.SL.TZ1.T_1e:
Show that Farmer Brown is incorrect.
17M.2.SL.TZ1.T_2d:
Find the length of the perimeter of ABCDE.
18N.2.SL.TZ0.T_5a:
Show that the size of angle BÂC is 20.2°, correct to 3 significant figures.
18M.2.SL.TZ2.T_5a.i:
Use Giovanni’s diagram to show that angle ABC, the angle at which the Tower is leaning relative to the
horizontal, is 85° to the nearest degree.
18M.2.SL.TZ2.T_5a.ii:
Use Giovanni's diagram to calculate the length of AX.
18M.2.SL.TZ2.T_5a.iii:
Use Giovanni's diagram to find the length of BX, the horizontal displacement of the Tower.
18M.2.SL.TZ2.T_5b:
Find the percentage error on Giovanni’s diagram.
17N.2.SL.TZ0.T_3a:
Show that ${\text{BD}} = 93{\text{ m}}$ correct to the nearest metre.
17N.2.SL.TZ0.T_3b:
Calculate angle ${\rm{B\hat CD}}$ .
17N.2.SL.TZ0.T_3d.i:
Calculate Abdallah’s estimate for the area.
17N.2.SL.TZ0.T_3d.ii:
Find the percentage error in Abdallah’s estimate.
19M.2.SL.TZ1.T_2a:
Find the length ${\text{AB}}$ , giving your answer to four significant figures.
19M.2.SL.TZ1.T_2b:
Find the total area of the two rectangles that make up the roof.
19M.2.SL.TZ1.T_2d:
Find the percentage error in her estimate.
19M.2.SL.TZ1.T_2e:
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State whether this new arrangement will allow Olivia to fit more solar panels to the roof. Justify your answer.
19M.2.SL.TZ2.T_2a:
Find the distance from A to C.
19M.2.SL.TZ2.T_2b.i:
Find the size of angle ${\text{B}}\mathop {\text{A}}\limits^ \wedge {\text{C}}$ .
19M.2.SL.TZ2.T_2b.ii:
Find the size of angle ${\text{C}}\mathop {\text{A}}\limits^ \wedge {\text{D}}$ .
19M.2.SL.TZ2.T_2c:
Find the size of angle ${\text{A}}\mathop {\text{C}}\limits^ \wedge {\text{D}}$ .
18M.2.SL.TZ1.T_1a:
Calculate the area of triangle EAD.
18M.2.SL.TZ1.T_1b:
Calculate the total volume of the barn.
18M.2.SL.TZ1.T_1f:
Calculate the total length of metal required for one support.
17M.2.SL.TZ1.T_2a.ii:
Show that the equation in part (a)(i) simplifies to $3{x^2} + 19x - 414 = 0$ .
17M.2.SL.TZ1.T_2b:
Find the length of CD.
17M.2.SL.TZ1.T_2e:
Calculate the length of CF.
18N.2.SL.TZ0.T_5d:
A vertical pole, TB, is constructed at point B and has height 25 m.

Calculate the angle of elevation of T from, M, the midpoint of the side AC.
17M.2.SL.TZ2.T_4a:
Find the length of the path BD.
17M.2.SL.TZ2.T_4c:
Find the area of the park.
16N.2.SL.TZ0.T_5c:
the area of triangle ABD;
16N.2.SL.TZ0.T_5e:
the length of AP;
20N.2.SL.TZ0.T_3a:
Calculate the length of $BD$ .
20N.2.SL.TZ0.T_3c:
Calculate the area of triangle $BDC$ .

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Markscheme

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